Question: A large brine tank containing a solution of salt and water is being diluted with fresh water. The relationship between the elapsed time, $t$, in hours, after the dilution begins, and the concentration of salt in the tank, $S(t)$, in grams per liter $(\text{g/l})$, is modeled by the following function. $ S(t)=600\cdot e^{{-0.3t}}$ How many hours will it take for the concentration of salt to decrease to $100\text{ g/l}$ ? Round your answer, if necessary, to the nearest hundredth.
Solution: Thinking about the problem We want to know how many hours, $t$, it will take for the concentration of salt, $S(t)$, to decrease to $100\text{ g/l}$. So we need to find the value of $t$ for which $S(t)=100$. Substituting $100$ in for $S(t)$ in the function gives us the following equation. $100=600\cdot e^{{-0.3t}}$ Solving the equation We can solve the equation as shown below. $\begin{aligned}600\cdot e^{-0.3t}&=100\\\\ e^{-0.3t}&=\dfrac{1}{6}\\\\\\ -0.3t&=\ln\left(\dfrac{1}{6}\right)\\\\\\ t&=\dfrac{\ln\left(\dfrac{1}{6}\right)}{-0.3}\\\\ t&\approx 5.97\end{aligned}$ The concentration will be $100\text{ g/l}$ after $5.97$ hours.